A short note on the rationality of the false
consensus effect
Christoph Vanberg∗
May 30, 2008
THIS NOTE IS WORK IN PROGRESS.
Abstract
In experiments which measure subjects’ beliefs, both beliefs about others’
behavior and beliefs about others’ beliefs, are often correlated with a subject’s own choices. Such phenomena have been interpreted as evidence of a
causal relationship between beliefs and behavior. An alternative explanation
attributes them to what psychologists refer to as a ‘false consensus effect.’ It
is my impression that the latter explanation is often prematurely dismissed
because it is thought to be based on an implausible psychological bias. The
goal of this note is to show that the false consensus effect does not rely on
such a bias. I demonstrate that rational belief formation implies a correlation of behavior and beliefs of all orders whenever behaviorally relevant traits
are drawn from an unknown common distribution. Thus, if we assume that
subjects rationally update beliefs, correlations of beliefs and behavior cannot
support a causal relationship.
KEYWORDS: Beliefs, behavioral economics, experimental economics
1
Max Planck Institute of Economics, Jena, Germany. Tel: +49 3641 686 636; Email: van-
[email protected]
1
Introduction
This short note is based on what used to be an Appendix to Vanberg (2008)1 .
My aim is to demonstrate that experimental economists should be careful when
lending a causal interpretation to observed correlations of beliefs and behavior. An
alternative interpretation of such a correlation is what psychologists refer to as the
‘false consensus effect’: People may systematically over-estimate the extent to which
others behave and think as they do.
It is my impression that this alternative explanation has not received enough
attention in the experimental literature seeking to investigate ostensibly causal relationships between beliefs and behavior. A possible reason is that economists think
that a false consensus effect would have to be based on a bizarre psychological bias,
and therefore the problem may be safe to ignore.
The short note below demonstrates, using a simple example, that this is wrong.
The false consensus effect is not based on any kind of psychological bias. If traits
relevant to behavior are correlated (drawn from the same unknown distribution), a
rational agent should use his own inclinations to predict the behavior of others, his
own beliefs to predict the beliefs of others, etc. Absent additional information, it
makes sense for people to hypothesize that others behave and think as they do.
At the moment, this is still just a short note based on a simple example. I believe
the argument can easily be generalized to a more general setup, and I intend to do
this in the future.
It has been brought to my attention that the argument I am formalizing here
has previously been made by Dawes (1989).2 I hope that the relatively simple
formalization I provide still has some added value.
1
Vanberg, C. (2008): Why do people keep their promises? An experimental test of two expla-
nations. Forthcoming in Econometrica
2
Dawes, R. (1989): Statistical Criteria for Establishing a Truly False Consensus Effect, Journal
of Experimental Social Psychology, 25, 1(17).
1
2
Model
Consider a world with two states labeled θ ∈ {θL , θH }, both equally likely. There
are N ≥ 2 players, who each have two available actions, aL and aH . Each player i
receives a private signal si ∈ {sL , sH }. In state θK , the probability that si = sK is
equal to p > 12 . Thus, each agent’s private signal is correlated with the state of the
world, which is common to all agents. Assume that behavior is entirely determined
by an agent’s signal. Specifically, when si = sK , agent i takes action aK .3
By construction, behavior in this example is not a function of an agent’s second
order beliefs. None the less, it is easy to show that second order beliefs will be
perfectly correlated with behavior. To see this, note first that a player who receives
signal sK attaches probability
1
p
2
1
1
p+ 2 (1−p)
2
=p>
1
2
to state θK . Thus, the posterior
probability that another agent j 6= i receives the same signal sK is given by q =
p2 + (1 − p)2 > 12 .
It follows that an agent who receives signal sK first order believes that another
agent will take action aK with probability µ1 (aK |sK ) = q > 21 . Now consider agent
i’s second order beliefs after receiving signal sK . With probability q, agent j 6= i
receives the same signal sK and (first order) believes that agent i will take action
aK with probability µ1 (aK |sK ) = q. With probability (1 − q), agent j receives
signal s−K and believes that i will take action aK with probability µ1 (aK |s−K ) =
(1 − q). Thus i second order believes that j attaches, in expectation, a probability
µ2 (aK |sK ) = q 2 +(1−q)2 >
1
2
to her (i) choosing action aK . Similarly, i believes that,
in expectation, j attaches probability µ2 (a−K |sK ) = 2 · q · (1 − q) = 1 − µ2 (aK |sK )
to her choosing action a−K .4
3
The signal can be interpreted in any number of ways. It may reflect a player’s type in terms of
intrinsic motivations to choose an action, or it may reflect information concerning the state of the
world, on which action preferences depend. What’s important is that the signal causes the agent
to behave in one way or the other.
4
With probability q, agent j believes that i will choose a−K with probability (1 − q). With
probability (1 − q), j attaches probability q to this event.
2
Now, consider what will happen in state θK . Suppose that we can observe behavior as well as (mean) second order beliefs concerning the probability of choosing
action aK . Clearly, an expected fraction p of all agents will choose action aK and
have second order beliefs µ2 (aK |sK ) > 21 . Conversely, an expected fraction (1 − p)
of all agents will choose action a−K and have second order beliefs µ2 (aK |s−K ) < 21 .
Thus, behavior will be perfectly correlated with second order beliefs even though it
is causally determined only by the si .
This example shows that a rational agent’s second order beliefs will tend to be
correlated with her behavior if private factors relevant to choice (e.g. preferences) are
correlated across agents. Thus, if experimental subjects believe that other subjects’
private preferences and inclinations are similar to their own, we should expect to
see a correlation of second order beliefs and behavior in any experimental setting,
even if behavior is driven by other factors.
3
Extension: Treatment effects
This example can be expanded to discuss the effects of a treatment variable on beliefs
and behavior. In addition to the private signals si , all agents now observe a public
signal t ∈ {0, 1}. Suppose that this signal directly affects the behavior of some
subjects. If t = 0, behavior is determined as before. If t = 1, a fraction r ∈ (0, 1]
of all agents prefers action aH , irrespective of their private signal. The remaining
‘flexible’ agents behave as before.
When t = 0, beliefs are determined as above. What happens to beliefs when
t = 1? An agent that receives signal sH will first order believe that others will
choose action aH with probability µ̃1 (aH |sH ) = r + (1 − r) · q > q = µ1 (aH |sH ). An
agent who receives signal sL will first order believe that others will choose action aH
with probability µ̃1 (aH |sL ) = r + (1 − r) · (1 − q) > (1 − q) = µ1 (aH |sL ). An agent
who receives signal sK will second order believe that another agent’s first order belief
is given by µ̃1 (aH |sK ) with probability q, and µ̃1 (aH |s−K ) with probability (1 − q).
In expectation, she believes that another agent attaches probability µ̃2 (aH |sK ) =
3
q · µ̃1 (aH |sK ) + (1 − q) · µ̃1 (aH |s−K ) > µ2 (aH |sK ) to the event that she will choose
action aH . Thus, both first and second order beliefs of all agents will put more
weight on action aH under the treatment condition.
Again, we can consider what would happen if we were to observe behavior and
beliefs in this setting. Clearly, nothing changes relative to the previous example
when t = 0. When t = 1 and θ = θH , an expected fraction p + r · (1 − p) of
all agents will choose action aH . (All those who receive signal sH , plus those who
receive signal sL , but are sensitive to the treatment.) Among these agents, the mean
second order belief will be β (aH , θH ) =
p·µ̃2 (aH |sH )+r·(1−p)·µ̃2 (aH |sL )
.
p+r·(1−p)
When θ = θL , an
expected fraction (1 − p) + r · p will choose action aH , and the mean second order
belief among these agents will be β (aH , θL ) =
(1−p)·µ̃2 (aH |sH )+r·p·µ̃2 (aH |sL )
.
(1−p)+r·p
Among
those choosing aL , the mean second order belief associated with action aH is equal
to β (aL , θK ) = µ̃2 (aH |sL ).
Relative to the baseline condition t = 0, the treatment condition t = 1 causes
the expected fraction of subjects choosing action aH to increase by r · (1 − p) when
θ = θH , and by r · p when θ = θL . Further, β (aH , θK ) > β (aL , θK ) for K = L, H.
That is, subjects choosing action aH will have ‘higher’ second order beliefs than
those choosing action aL .
Thus, the data will have the following features: (1) beliefs and behavior are correlated within each of the treatment conditions (2) second order beliefs are correlated
with the treatment condition, and (3) behavior is correlated with the treatment
condition. Despite the fact that behavior is directly affected by the treatment signal
t, features (2) and (3) are consistent with the hypothesis that behavior is causally
driven by second order beliefs.
4